Péter Gáspár, Zoltán Szabó, József Bokor

Discrete Feedback Systems 2.

Modern Control


A.1.3 Schur complement and Schur lemma

Lemma A.1 (Schur Decomposition) Suppose A or D respectively is non non-singular. Then
 
, or
 
 
Lemma A.2 (Schur Lemma) Let and be symmetric matrices. The following are equivalent.
 
(A.16)
(A.17)
(A.18)
 
 
 
Lemma A.3 (Symmetric Schur Lemma) Let and be symmetric matrices. The following are equivalent.
 
(A.19)
(A.20)
(A.21)
 
We note that the equality is redundant since .
 
 
Lemma A.4 Suppose that is nonsingular. Then
 
 
 
Lemma A.5 (Matrix Inversion Lemma) Let , and be non-singular. Then
 
 
Suppose and are both non-singular. Then
 
 
Let us suppose that the block matrix
 
 
is non-singular. Moreover, suppose or respectively is non-singular and let or . Then
 
or
 
 
If , and are all non-singular then
 
 
An easy but important observation is that for the quadratic form we have
 
 
i.e., we have the following sum of squares representations
 
(A.22)
 
if exists and
 
(A.23)
 
if exists, respectively. As an example, having the problem under conditions
 
 
we have immediate the conditions , getting . Impozing , we can obtain the lowest value, i.e., .
 

Discrete Feedback Systems 2.

Tartalomjegyzék


Kiadó: Akadémiai Kiadó

Online megjelenés éve: 2019

ISBN: 978 963 454 373 2

The classical control theory and methods that we have been presented in the first volume are based on a simple input-output description of the plant, expressed as a transfer function, limiting the design to single-input single-output systems and allowing only limited control of the closed-loop behaviour when feedback control is used. Typically, the need to use modern linear control arises when working with models which are complex, multiple input multiple output, or when optimization of performance is a concern. Modern control theory revolves around the so-called state-space description. The state variable representation of dynamic systems is the basis of different and very direct approaches applicable to the analysis and design of a wide range of practical control problems. To complete the design workflow, finally some introduction into system identification theory is given.

Hivatkozás: https://mersz.hu/gaspar-szabo-bokor-discrete-feedback-systems-2//

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